1 Suppose f x y has values and partial derivatives as in the table at right Find all the critical points you can from the given data and classify them into local mins local maxes and saddles 3 points x y f fx fy f xx fx y fyy 0 0 1 0 2 1 0 2 5 0 1 0 0 1 0 2 1 1 1 0 3 3 2 2 Local mins if any Local maxes if any Saddles if any 2 Find an equation of the tangent plane to the surface defined by 3x 2 x y 2y z 8 at the point 1 1 2 3 points Equation x y z 3 Let f x y x y 1 Let C be the curve defined by x 2 4y 2 8 a Find the maximum value M and minimum value m achieved by f x y on the curve C 5 points M m b The curve C is shown below On the same plot graph the level set f x y M 2 points 2 1 3 2 1 1 2 3 1 2 c What two properties does the curve C have that guarantee f x y has absolute maximum and minimum values on C 1 point each 4 Consider the space curve C parameterized by r t cos t 2 sin t cos t for 0 t 2 a Mark the correct sketch of C below 2 points z z z x x x y b Evaluate the line integral y C y z 1 ds 5 points C y z 1 ds y 5 Consider the vector field F x y y a Mark the picture of F below 2 points 2 b Consider the curve C parameterized by r t t t for 0 t 1 Compute F d r 4 points F dr C C 6 Find a parameterization r t of the curve of intersection of the paraboloid y 2x 2 z 2 and the cylinder x 2 z 2 1 3 points r t 7 a Consider the vector field F x y y 2 1 2x y on R2 Show that F is conservative by finding a function f x y where F f 3 points f x y b For the curve C shown at right evaluate C F d r 2 points 2 1 2 1 1 2 1 C F dr c Exactly one of the vector fields below is not conservative Mark the box of the non conservative vector field 2 point 8 Consider the function f R2 R given by the contour diagram at right as well as the curve C the point A and the vectors u and v indicated For each part circle the best answer 1 point each a The sign of D u f A negative zero f 7 positive f 5 f 4 3 b f A v true f 6 A false f 3 u c The value of 9 6 C f x y ds 3 0 3 2 6 f 2 v C 9 f 1 1 0 0 1 2 3 9 Let g R2 R be the function whose graph is shown at right and let C be the indicated curve in the x y plane Evaluate the line integral C g d r 2 points z g x y z z 30 z 20 z 10 y C C g d r x
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